114	5 Cosmology and General Relativity
where
	η = diag(+, +, . . . , +, −)	(5.2.21)

This guarantees that L(y) are elements of SO(n, 1), secondly observe that the image x(y) of the standard vector x0 through L(y),

			0								2y1
			.							.
x(y) L(y)x0			.				1		2			.				(5.2.22)
		L(y) .									2yn
≡	=		0		=	1		y				.
							−
											1	+	y2
			1										2
											1−y
lies, as it should, in the algebraic locus H−(n,1),
	x(y)T ηx(y) = −1															(5.2.23)

and has n linearly independent entries (the first n) parameterized by y. Hence the lateral classes can be labeled by y and this concludes our argument to show that (5.2.19) is a good coset parameterization. L(0) = 1(n+1)×(n+1) corresponds to the identity class which is usually named the origin of the coset.

5.2.3 The Geometry of Coset Manifolds

In order to study the geometry of a coset manifold G/H, the first important step is provided by the orthogonal decomposition of the corresponding Lie algebra, namely by

G = H K

(5.2.24)

where G is the Lie algebra of G and the subalgebra H G is the Lie algebra of the subgroup H and where K denotes a vector space orthogonal to H with respect to the Cartan Killing metric of G. By definition of subalgebra we always have:

[H, H] H	(5.2.25)
while in general one has:
[H, K] H K	(5.2.26)

Definition 5.2.3 Let G/H be a Lie coset manifold and let the orthogonal decomposition of the corresponding Lie algebra be as in (5.2.24). If the condition:

[H, K] K

(5.2.27)

applies, the coset G/H is named reductive.

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115

Equation (5.2.27) has an obvious and immediate interpretation. The complementary space K forms a linear representation of the subalgebra H under its adjoint action within the ambient algebra G.

Almost all of the “reasonable” coset manifolds which occur in various provinces of Mathematical Physics are reductive. Violation of reductivity is a sort of pathology whose study we can disregard in the scope of this book. We will consider only reductive coset manifolds.

Definition 5.2.4 Let G/H be a reductive coset manifold. If in addition to (5.2.27) also the following condition:

[K, K] H

(5.2.28)

applies, then the coset manifold G/H is named a symmetric space.

Let TA (A = 1, . . . , n) denote a complete basis of generators for the Lie algebra G:

[TA, TB ] = CCAB TC

(5.2.29)

and Ti (i = 1, . . . , m) denote a complete basis for the subalgebra H G. We also introduce the notation Ta (a = 1, . . . , n − m) for a set of generators that provide a basis of the complementary subspace K in the orthogonal decomposition (5.2.24). We nickname Ta the coset generators. Using such notations, (5.2.29) splits into the following three ones:

[Tj , Tk ] = Cij k Ti	(5.2.30)
[Ti , Tb] = Caib Ta	(5.2.31)
[Tb, Tc] = Cibc Ti + Cabc Ta	(5.2.32)
Equation (5.2.30) encodes the property of H of being a	subalgebra. Equa-

tion (5.2.31) encodes the property of the considered coset of being reductive. Finally if in (5.2.32) we have Cabc = 0, the coset is not only reductive but also symmetric.

We will be able to provide explicit formulae for the Riemann tensor of reductive coset manifolds equipped with G-invariant metrics in terms of such structure constants. Prior to that we consider the infinitesimal transformation and the very definition of the Killing vectors with respect to which the metric has to be invariant.

5.2.3.1 Infinitesimal Transformations and Killing Vectors

Let us consider the transformation law (5.2.17) of the coset representative. For a group element g infinitesimally close to the identity, we have:

g 1	+ εATA	(5.2.33)
h(y, g) 1	− εAWAi (y)Ti	(5.2.34)
y α yα + εAkAα		(5.2.35)

116	5 Cosmology and General Relativity

The induced h transformation in (5.2.17) depends in general on the infinitesimal G-parameters εA and on the point in the coset manifold y, as shown in (5.2.34). The y-dependent rectangular matrix WAi (y) is usually named the H-compensator. The shift in the coordinates yα is also proportional to εA and the vector fields:

kA = kAα	∂	(5.2.36)
	(y) ∂yα

are named the Killing vectors of the coset. The reason for such a name will be justified when we will show that on G/H we can construct a (pseudo-)Riemannian metric which admits the vector fields (5.2.36) as generators of infinitesimal isometries. For the time being those in (5.2.36) are just a set of vector fields that, as we prove few lines below, close the Lie algebra of the group G.

Inserting (5.2.33)–(5.2.35) into the transformation law (5.2.17) we obtain:

TAL(y) = kAL(y) − WAi (y)L(y)Ti

(5.2.37)

Consider now the commutator g2−1g1−1g2g1 acting on L(y). If both group elements g1,2 are infinitesimally close to the identity in the sense of (5.2.33), then we obtain:

g2−1g1−1g2g1L(y) 1 − ε1Aε2B [TA, TB ] L(y)

(5.2.38)

By explicit calculation we find:

[TA, TB ]L(y) = TATB L(y) − TB TAL(y)

= [kA, kB ]L(y) − kAWBi − kB WAi + 2Cij k WAj WBk L(y)Ti

(5.2.39)

On the other hand, using the Lie algebra commutation relations we obtain:

[TA, TB ]L(y) = CCAB TC L(y) = CCAB kC L(y) − WCi L(y)Ti

(5.2.40)

By equating the right hand sides of (5.2.39) and (5.2.40) we conclude that:

[kA, kB ] = CCAB kC	(5.2.41)
kAWBi − kB WAi + 2Cij k WAj WBk = CCAB WCi	(5.2.42)

where we separately compared the terms with and without W’s, since the decomposition of a group element into L(y)h is unique.

Equation (5.2.41) shows that the Killing vector fields defined above close the commutation relations of the G-algebra.

Equation (5.2.42) will be used to construct a consistent H-covariant Lie derivative.

In the case of the spaces H(n,− 1), which we choose as illustrative example, the Killing vectors can be easily calculated by following the above described procedure

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117

step by step. For later purposes we find it convenient to present such a calculation in a slightly more general set up by introducing the following coset representative that depends on a discrete parameter κ = ±1:

											1n×n + 2yyT								κ				−2		y
						Lκ (y)		=										1+κy2						1+κy2								(5.2.43)
																							1+κy2
																1+κy2
														2κ		yT							1−κy2
An explicit calculation shows that:
			0			1	· · ·		0		0										0		1	· · ·			0		0
					1	0	· · ·		0		0										1		0	· · ·			0		0
	κ (y)										.				κ (y)														.			(5.2.44)
			. . . .														=				. . . .
L		T			. . . .						.			L							. . . .								.
					. . . .						.										. . . .								.
					0		0 1				0										0			0 1					0
						· · ·																	· · ·
					0		0 0				κ										0			0 0					κ
						· · ·																	· · ·
							ηκ																		ηκ

Namely L−1(y) is an SO(n, 1) matrix, while L1(y) is an SO(n + 1) group element. Furthermore defining, as in (5.2.22):

				0
				.
				.
xκ (y)	≡	Lκ (y)		.		(5.2.45)
				0
				1
we find that:
xκ (y)T ηκ xκ (y) = κ						(5.2.46)

Hence by means of L1(y) we parameterize the points of the n-sphere Sn, while by means of L−1(y) we parameterize the points of H(n,− 1) named also the n-pseudo- sphere or the n-hyperboloid. In both cases the stability subalgebra is so(n) for which a basis of generators is provided by the following matrices:

	Jij		= Iij − Ij i ;		i, j = 1, . . . , n				(5.2.47)
having named:
		0	· · ·	· ·	0		0
		0	· · ·	·1	0		0 } ith row
	=	0	· · ·	· · ·	0		0		(5.2.48)
Iij		0			0		0	.
			· · ·	· · ·
				j th column

the (n + 1) × (n + 1) matrices whose only non-vanishing entry is the ij th one, equal to 1.